Graphs on surfaces and Khovanov homology

Abhijit Champanerkar; Ilya Kofman; Neal Stoltzfus

doi:10.2140/agt.2007.7.1531

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arXiv · 0705.3453

Graphs on surfaces and Khovanov homology

Abhijit Champanerkar·Ilya Kofman·Neal Stoltzfus

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Abstract

Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We show that for any link diagram $L$, there is an associated ribbon graph whose quasi-trees correspond bijectively to spanning trees of the graph obtained by checkerboard coloring $L$. This correspondence preserves the bigrading used for the spanning tree model of Khovanov homology, whose Euler characteristic is the Jones polynomial of $L$. Thus, Khovanov homology can be expressed in terms of ribbon graphs, with generators given by ordered chord diagrams.

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BibTeX RIS

Abhijit Champanerkar, Ilya Kofman, Neal Stoltzfus. 2007-05-23. Graphs on surfaces and Khovanov homology. https://doi.org/10.2140/agt.2007.7.1531

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